The arithmetic of the product of two algebraic curves over a finite field

Abstract

Let X = C 1 × C 2 be the product of two nonsingular projective curves defined over a finite field k = F q . We compute the order of the Brauer group Br( X) of X. In the generic case, it is given by the resultant of the characteristic polynomials of the Frobenius endomorphism of the Jacobian variety J( C i ) ( i = 1, 2), divided by a certain power of q. In particular, if at least one component C i has ordinary Jacobian variety J( C i ), then the order of Br( X) is equal to the resultant divided by q p g , where p g is the geometric genus of X.